The gaps left by a Brownian motion

Run a Brownian motion on a torus for a long time. How large are the random gaps left behind when the path is removed?

In three (or more) dimensions, we find that there is a deterministic spatial scale common to all the large gaps anywhere in the torus. Moreover, we can identify whether a gap of a given shape is likely to exist on this scale, in terms of a single parameter, the classical (Newtonian) capacity. I will describe why this allows us to identify a well-defined "component" structure in our random porous set.